Taranovsky's interesting recent note [Feb 27, 2005] on extensions of set
theory has inspired me to briefly discuss some recent results that
characterize the rather surprising strength of a natural system of set
theory with a class of indiscernibles, here dubbed ZFCI.
ZFCI is a theory formulated in the language {epsilon, I(x), <}, where I(x)
is a unary predicate [to distinguish the indiscernibles] and < is a binary
relation [for a global well-ordering]. Intuitively speaking, ZFCI is an
extension of ZFC that strongly negates Leibniz's dictum on the identity of
indiscernibles by asserting "there are a proper class of indiscernibles".
The axioms of ZFCI are as follows:
0. The axioms of ZFC;
1. the sentence expressing "I is a proper class of ordinals";
2. a schema expressing that < is a global well-ordering;
3. the Replacement scheme for formulae using I and <; and
4. the Indiscernibility scheme, which is a scheme asserting that (I,<) is a
class of order indiscernibles [in the usual sense of model theory] for
formulae in the language {epsilon, <}.
It turns out that ZFCI goes well beyond ZFC since it proves the existence of
n-Mahlo cardinals for each concrete natural number n. However, if
consistent, it will not prove the statement "for each natural number n,
there is an n-Mahlo cardinal".
Indeed, one can precisely describe the first order consequences of ZFCI in
the usual language of set theory {epsilon}. In order to do so, let PHI be
the set of sentences of following form (where n is a concrete natural
number):
"there is an n-Mahlo cardinal kappa such that V(kappa) is a SIGMA_n
elementary submodel of the universe".
It is not hard to see that ZFC + PHI is equiconsistent with ZFC plus axioms
of the form "there is an n-Mahlo cardinal" (again, where n is a concrete
natural number n). The former theory of course proves the latter theory, but
not vice versa.
Here is the first main result:
Theorem A. For any sentence S in the usual language of set theory {epsilon},
the following two conditions are equivalent:
(i) ZFCI proves S;
(ii) ZFC + PHI proves S.
It is worth pointing out that the situation for Peano arithmetic (or
equivalently: the theory of finite sets) is quite different, i.e., if one
formulates an analogous theory, PAI, extending PA, then the formalized
version of Ramsey's theorem can be used to show that PAI proves precisely
the same arithmetical sentences as PA itself. This adds more plausibility to
the theory ZFCI since it shows that the indiscernibles allow ZFC to
"catch-up" with PA.
One may also wish to *iterate* the idea of adding indiscernibles by adding
countably many new unary predicates I_n (x) for each natural number n in
order to formulate a theory ZFCI# extending ZFCI by adding axioms asserting
that I_(n+1) is a proper class of indiscernibles for formulae in the
language {epsilon, <}augmented with I_1, ., I_n.
As it turns out, this will not buy us any new theorem of set theory that
ZFCI could not prove already, i.e., Theorem A can be improved to the
following result which connects ZFCI and ZFCI! to other systems of set
theory.
Theorem B. For any sentence S in the usual language of set theory {epsilon},
the following five conditions are equivalent:
(i) ZFCI# proves S;
(ii) ZFCI proves S;
(iii) GBC + "the class of ordinals is weakly compact" proves S;
(iv) NFUA proves "S holds in CZ";
(v) ZFC + PHI proves S.
Here GBC is the Godel-Bernays theory of classes; NFUA is the extension of
the Quine-Jensen system of set theory NFU with a universal set obtained by
adding the axioms of Choice, Infinity, and the axiom expressing "every
Cantorian set is strongly Cantorian"; and CZ is the canonical model of ZFC
that can be *interpreted* in models of NFUA.
The equivalence of (i), (ii) and (v) is will be soon available, but the
equivalence of (iii), (iv), and (v) [which were inspired by the work of
Solovay and Holmes] appear in my paper:
Automorphisms, Mahlo Cardinals, and NFU, Nonstandard Models of Arithmetic
and Set Theory, Contemporary Mathematics (Enayat and Kossak, ed.), volume
361, American Mathematical Society, 2004.
Also available on: http://academic2.american.edu/~enayat/Aut.pdf
Best regards,
Ali Enayat